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Chapter 10. Gases
10.1 Characteristics of Gases
• All substances have three phases: solid, liquid and gas.
• Substances that are liquids or solids under ordinary conditions may also exist as gases.
• These are often referred to as vapors.
• Many of the properties of gases differ from those of solids and liquids:
• Gases are highly compressible and occupy the full volume of their containers.
• When a gas is subjected to pressure, its volume decreases.
• Gases always form homogeneous mixtures with other gases.
• Gases only occupy a small fraction of the total volume of their containers.
• As a result, each molecule of gas behaves largely as though other molecules were absent.
10.2 Pressure
• Pressure is the force acting on an object per unit area:
F
P
A
Atmospheric Pressure and the Barometer
• The SI unit of force is the newton (N).
• 1 N = 1 kgm/s2
• The SI unit of pressure is the pascal (Pa).
• 1 Pa = 1 N/m2
• A related unit is the bar, which is equal to 105 Pa.
• Gravity exerts a force on the Earth’s atmosphere.
• A column of air 1 m2 in cross section extending to the upper atmosphere exerts a force of
5
10 N.
• Thus, the pressure of a 1 m2 column of air extending to the upper atmosphere is 100 kPa.
• Atmospheric pressure at sea level is about 100 kPa or 1 bar.
• The actual atmospheric pressure at a specific location depends
on the altitude and the weather conditions.
• Atmospheric pressure is measured with a barometer.
• If a tube is completely filled with mercury and then inverted into a container of mercury
open to the atmosphere, the mercury will rise 760 mm up the tube.
• Standard atmospheric pressure is the pressure required to support 760 mm of Hg in a
column.
• Important non SI units used to express gas pressure include:
• atmospheres (atm)
• millimeters of mercury (mm Hg) or torr
• 1 atm = 760 mm Hg = 760 torr = 1.01325 x 105 Pa = 101.325 kPa
10.3 The Gas Laws
• The equations that express the relationships among T (temperature), P (pressure), V (volume), and n
(number of moles of gas) are known as the gas laws.
The Pressure-Volume Relationship: Boyle’s Law
• Weather balloons are used as a practical application of the relationship between pressure and volume
of a gas.
• As the weather balloon ascends, the volume increases.
• As the weather balloon gets further from Earth’s surface, the atmospheric pressure
decreases.
• Boyle’s law: The volume of a fixed quantity of gas, at constant temperature, is inversely proportional
to its pressure.
• Mathematically:
1
V constan
t or PV constant
P
• A plot of V versus P is a hyperbola.
V
V constant T or constant
T
• A plot of V versus 1/P must be a straight line passing through the origin.
• The working of the lungs illustrates that:
• as we breathe in, the diaphragm moves down, and the ribs expand; therefore, the volume
of the lungs increases.
• according to Boyle’s law, when the volume of the lungs increases, the pressure decreases;
therefore, the pressure inside the lungs is less than the atmospheric pressure.
• atmospheric pressure forces air into the lungs until the pressure once again equals
atmospheric pressure.
• as we breathe out, the diaphragm moves up and the ribs contract; therefore, the volume of
the lungs decreases.
• By Boyle’s law, the pressure increases and air is forced out.
The Temperature-Volume Relationship: Charles’s Law
• We know that hot-air balloons expand when they are heated.
• Charles’s law: The volume of a fixed quantity of gas at constant pressure is directly proportional to its
absolute temperature.
• Mathematically:
• Note that the value of the constant depends on the pressure and the number of moles of
gas.
• A plot of V versus T is a straight line.
• When T is measured in C, the intercept on the temperature axis is –273.15C.
• We define absolute zero, 0 K = –273.15C.
The Quantity-Volume Relationship: Avogadro’s Law
• Gay-Lussac’s law of combining volumes: At a given temperature and pressure the volumes of gases
that react with one another are ratios of small whole numbers.
• Avogadro’s hypothesis: Equal volumes of gases at the same temperature and pressure contain the
same number of molecules.
• Avogadro’s law: The volume of gas at a given temperature and pressure is directly proportional to the
number of moles of gas.
• Mathematically:
V = constant x n
• We can show that 22.4 L of any gas at 0C and 1 atmosphere contains 6.02 x 1023 gas
molecules.
10.4 The Ideal-Gas Equation
• Summarizing the gas laws:
• Boyle: V 1/P (constant n, T)
• Charles: V T (constant n, P)
• Avogadro: V n (constant P, T)
• Combined: V nT/P
• Ideal gas equation: PV = nRT
• An ideal gas is a hypothetical gas whose P, V, and T behavior is completely described by
the ideal-gas equation.
• R = gas constant = 0.08206 Latm/molK
• Other numerical values of R in various units are given in Table 10.2.
• Define STP (standard temperature and pressure) = 0C, 273.15 K, 1 atm.
• The molar volume of 1 mol of an ideal gas at STP is 22.41 L.
Relating the Ideal-Gas Equation and the Gas Laws
• If PV = nRT and n and T are constant, then PV is constant and we have Boyle’s law.
• Other laws can be generated similarly.
• In general, if we have a gas under two sets of conditions, then
PV1 P2V2
1
n1T1 n2T2
• We often have a situation in which P, V, and T all change for a fixed number of moles of gas.
• For this set of circumstances,
PV
nR constant
T
• Which gives
P1V1 P2V2
T1 T2
10.5 Further Applications of the Ideal-Gas Equation
Gas Densities and Molar Mass
• Density has units of mass over volume.
• Rearranging the ideal-gas equation with M as molar mass we get
n P
V RT
nM PM
V RT
PM
d
RT
• The molar mass of a gas can be determined as follows:
dRT
M
P
Volumes of Gases in Chemical Reactions
• The ideal-gas equation relates P, V, and T to number of moles of gas.
• The n can then be used in stoichiometric calculations.
10.6 Gas Mixtures and Partial Pressures
• Since gas molecules are so far apart, we can assume that they behave independently.
• Dalton observed:
• The total pressure of a mixture of gases equals the sum of the pressures that each
would exert if present alone.
• Partial pressure is the pressure exerted by a particular component of a gas mixture.
• Dalton’s law of partial pressures: In a gas mixture the total pressure is given by the sum of partial
pressures of each component:
Pt = P1 + P2 + P3 + …
• Each gas obeys the ideal gas equation.
• Thus,
RT RT
Pt (n1 n2 n3 ) nt
V V
Partial Pressures and Mole Fractions
• Let n be the number of moles of gas 1 exerting a partial pressure P1, then
1
P1 = Pt
• where is the mole fraction (n1/nt).
• Note that a mole fraction is a dimensionless number.
Collecting Gases over Water1
• It is common to synthesize gases and collect them by displacing a volume of water.
• To calculate the amount of gas produced, we need to correct for the partial pressure of the water:
Ptotal = Pgas + Pwater
• The vapor pressure of water varies with temperature.
• Values can be found in Appendix B.
10.7 Kinetic-Molecular Theory
• The kinetic-molecular theory was developed to explain gas behavior.
• It is a theory of moving molecules.
• Summary:
• Gases consist of a large number of molecules in constant random motion.
• The combined volume of all the molecules is negligible compared with the volume of the
container.
• Intermolecular forces (forces between gas molecules) are negligible.
• Energy can be transferred between molecules during collisions, but the average kinetic
energy is constant at constant temperature.
• The collisions are perfectly elastic.
• The average kinetic energy of the gas molecules is proportional to the absolute
temperature.
• Kinetic molecular theory gives us an understanding of pressure and temperature on the molecular
level.
• The pressure of a gas results from the collisions with the walls of the container.
• The magnitude of the pressure is determined by how often and how hard the molecules
strike.
• The absolute temperature of a gas is a measure of the average kinetic energy.
• Some molecules will have less kinetic energy or more kinetic energy than the
average
(distribution).
• There is a spread of individual energies of gas molecules in any sample of gas.
• As the temperature increases, the average kinetic energy of the gas molecules increases.
• As kinetic energy increases, the velocity of the gas molecules increases.
• Root-mean-square (rms) speed, u, is the speed of a gas molecule having average kinetic
energy.
• Average kinetic energy, , is related to rms speed:
= ½mu2
• where m = mass of the molecule.
Application to the Gas-Laws
• We can understand empirical observations of gas properties within the framework of the kinetic-
molecular theory.
• The effect of an increase in volume (at constant temperature) is as follows:
• As volume increases at constant temperature, the average kinetic energy of the gas
remains constant.
• Therefore, u is constant.
• However, volume increases, so the gas molecules have to travel further to hit the walls of
the container.
• Therefore, pressure decreases.
• The effect of an increase in temperature (at constant volume) is as follows:
• If temperature increases at constant volume, the average kinetic energy of the gas
molecules increases.
• There are more collisions with the container walls.
• Therefore, u increases.
• The change in momentum in each collision increases (molecules strike harder).
• Therefore, pressure increases.
10.8 Molecular Effusion and Diffusion
• The average kinetic energy of a gas is related to its mass:
= ½m 2
• Consider two gases at the same temperature: the lighter gas has a higher rms speed than the heavier
gas.
• Mathematically:
3RT
u
M
• The lower the molar mass, M, the higher the rms speed for that gas at a constant
temperature.
• Two consequences of the dependence of molecular speeds on mass are:
• Effusion is the escape of gas molecules through a tiny hole into an evacuated space.
• Diffusion is the spread of one substance throughout a space or throughout a second
substance.
Graham’s Law of Effusion
• The rate of effusion can be quantified.
• Consider two gases with molar masses, M1 and M2, and with effusion rates, r1 and r2, respectively.
• The relative rate of effusion is given by Graham’s law:
r1 M2
r2 M1
• Only those molecules which hit the small hole will escape through it.
• Therefore, the higher the rms speed the more likely it is that a gas molecule will hit the
hole.
• We can show
r1 u1 M2
r2 u 2 M1
Diffusion and Mean Free Path
• Diffusion is faster for light gas molecules.
• Diffusion is significantly slower than the rms speed.
• Diffusion is slowed by collisions of gas molecules with one another.
• Consider someone opening a perfume bottle: It takes awhile to detect the odor, but the
average speed of the molecules at 25C is about 515 m/s (1150 mi/hr).
• The average distance traveled by a gas molecule between collisions is called the mean free path.
• At sea level, the mean free path for air molecules is about 6 x 10 – 6 cm.
10.9 Real Gases: Deviations from Ideal Behavior
• From the ideal gas equation:
PV
n
RT
• For 1 mol of an ideal gas, PV/RT = 1 for all pressures.
• In a real gas, PV/RT varies from 1 significantly.
• The higher the pressure the more the deviation from ideal behavior.
• For 1 mol of an ideal gas, PV/RT = 1 for all temperatures.
• As temperature increases, the gases behave more ideally.
• The assumptions in the kinetic-molecular theory show where ideal gas behavior breaks down:
• The molecules of a gas have finite volume.
• Molecules of a gas do attract each other.
• As the pressure on a gas increases, the molecules are forced closer together.
• As the molecules get closer together, the free space in which the molecules can move gets
smaller.
• The smaller the container, the more of the total space the gas molecules occupy.
• Therefore, the higher the pressure, the less the gas resembles an ideal gas.
• As the gas molecules get closer together, the intermolecular distances decrease.
• The smaller the distance between gas molecules, the more likely that attractive forces will
develop between the molecules.
• Therefore, the less the gas resembles an ideal gas.
• As temperature increases, the gas molecules move faster and further apart.
• Also, higher temperatures mean more energy is available to break intermolecular forces.
• As temperature increases, the negative departure from ideal-gas behavior disappears.
The van der Waals Equation
• We add two terms to the ideal gas equation to correct for
• the volume of V nb molecules:
• for molecular attractions:
n2a
2
V
• The correction terms generate the van der Waals equation:
• where a and b are empirical constants that differ for each gas.
n2a
P 2 V nb nRT
V
• van der Waals constants for some common gases can be found in Table 10.3.
• To understand the effect of intermolecular forces on pressure, consider a molecule that is about to
strike the wall of the container.
• The striking molecule is attracted by neighboring molecules.
• Therefore, the impact on the wall is lessened.
